A Fixed-b Perspective on the Phillips-Perron Tests

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Summary

In this paper we extend fixed-b asymptotic theory to the nonparametric Phillips–
Perron (PP) unit root tests. We show that the fixed-b limits depend on nuisance
parameters in a complicated way. These nonpivotal limits provide an alternative
theoretical explanation for the well-known finite-sample problems of the PP tests.
We also show that the fixed-b limits depend on whether deterministic trends are
removed using one-step or two-step detrending approaches. This is in contrast to
the asymptotic equivalence of the one- and two-step approaches under a consistency
approximation for the long-run variance estimator. Based on these results we
introduce modified PP tests that allow for asymptotically pivotal fixed-b inference.
The theoretical analysis is cast in the framework of near-integrated processes, which
allows us to study the asymptotic behavior both under the unit root null hypothesis
and for local alternatives. The performance of the original and modified PP tests is
compared by means of local asymptotic power and a small finite-sample simulation
study.